I have a particular empirical discrete data set observing consumption characteristics of a certain population according to time. Plotted on a graph, it produces a vector field $\mathbb{N}^2 \to \mathbb{R}^2$ which looks like:
The data set actually has way more points than this, but its general shape is remarkably consistent (with some, but few, discrepancies) and looks like the above image.
I am wondering: this actually looks like the graph of some differential equation, if we ignore the fact the domain of the function is discrete. Is it at all possible to approximate this vector field using a differential equation, i.e., find the differential equation that would best reproduce this vector field (ignoring small discrepancies)? And if so, would anyone have recommendations on where I could start learning how to do this? I might also be very mistaken in believing this is possible or I might be misinterpreting the relevant mathematical concepts, so any pointers would be greatly appreciated. Thank you very much!
Best Answer
This goes under the name System identification. One can apply several kinds data-driven methods to recover the original dynamics.